Linearized optical digital-to-analog modulator

ABSTRACT

In a system for converting digital data into a modulated optical signal, an electrically controllable device, including a modulator having one or more actuating electrodes, provides an analog-modulated optical signal that is modulated in response to output data bits of a digital-to-digital mapping. A digital-to-digital conversion provides the mapping of input data words to the output data bits. The mapping enables adjustments to correct for non-linearities and other undesirable characteristics, thereby improving signal quality.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/481,904 filed on Sep. 22, 2021, which is a continuation of U.S.patent application Ser. No. 16/532,567 filed on Aug. 6, 2019, now U.S.Pat. No. 11,133,872, which is a continuation of U.S. patent applicationSer. No. 16/386,391 filed Apr. 17, 2019, now U.S. Pat. No. 10,461,866,which is a continuation of U.S. patent application Ser. No. 16/234,635filed on Dec. 28, 2018, now U.S. Pat. No. 10,270,535, which is acontinuation of U.S. patent application Ser. No. 15/298,373 filed onOct. 20, 2016, now U.S. Pat. No. 10,205,527, which is a continuation ofU.S. patent application Ser. No. 14/922,165 filed on Oct. 25, 2015, nowU.S. Pat. No. 9,479,191, which is a continuation of U.S. patentapplication Ser. No. 14/662,343 filed on Mar. 19, 2015, now U.S. Pat.No. 9,203,425, which is a continuation of U.S. patent application Ser.No. 14/325,486 filed on Jul. 8, 2014, now U.S. Pat. No. 9,031,417, whichis a continuation of U.S. patent application Ser. No. 13/280,371 filedon Oct. 25, 2011 now U.S. Pat. No. 8,797,198, which is a continuation ofU.S. patent application Ser. No. 12/636,805, filed on Dec. 14, 2009, nowU.S. Pat. No. 8,044,835, which is a continuation-in-part of PCT PatentApplication No. PCT/IL2008/000805 filed on Jun. 12, 2008, which claimsthe benefit of priority of U.S. Provisional Patent Application No.60/943,559 filed on Jun. 13, 2007. The contents of the aboveapplications are all incorporated by reference as if fully set forthherein in their entirety.

FIELD AND BACKGROUND OF THE INVENTION

The present invention relates to optical modulators and, in particular,it concerns a linearized optical digital-to-analog modulator.

There is a tangible need for high-performance and large bandwidthdigital to analog signal conversion. Furthermore, as the RF and digitaldomains converge, entirely new solutions will be needed to enablemulti-GHz mixed-signal systems. Probably the most prominent area tobenefit is the wireless communication industry. The ever increasingthirst for bandwidth will require data converters to deliver greatlyincreased performance. For example, analog signals are transmitted incable television (CATV) via optical fibers and the demand for increasingbandwidth is driving technology to speed-up the processing of signals aswell as the transmission. High performance digital to analog conversionis also required to address the growing demands of wireless carriers forsupporting the heavy traffic expected in the base station. Additionalspecific areas to benefit include: the defense and government industriesthat concentrate on deploying multi-function, dynamically reconfigurablesystems (RADAR, electronic warfare, and surveillance applications);medical imaging; and hyper/super-computer communications.

One of the most widely deployed devices for analog optics modulation isthe Mach-Zehnder Interferometer modulator (MZI). For binary digitalsignals, it is today the preferred device for long-haul fiber-opticcommunication, leading to chirp-free pulses which can reach hundreds ofkilometers in optical fibers without the need for regeneration. Foranalog applications, however, a serious problem is encountered due tothe inherent non-linear response of the modulator. Specifically, sincethe modulating voltage via the electro-optic effect controls the opticalphase delay in a basically linear fashion and the attenuation varies asthe cosine of the phase difference between the two branches of thedevice, a linear variation in phase difference and thus in appliedvoltage results in a cosine-shaped output variation, as seen in thepattern of points in FIG. 2A. The common solutions for this problem areeither the biasing of the device to a quasi linear regime coupled withreducing the modulation range to reduce distortion, or use of an analogpre-distortion circuit to feed the modulator. Since in practically allpresent systems signals are processed digitally, a multi-bitDigital-to-Analog Converter (DAC) device is needed with fast processingcapabilities.

A DAC based on a multi-electrode MZI modulator concept was proposed manyyears ago by Papuchon et al. and is described in U.S. Pat. No.4,288,785. In that device, the electrodes' sectioning length followed aconventional power-of-two digital sequence, which did not solve thenon-linearity problem, and thus suffered from severe limitation in thedynamic range, and subsequently the attainable resolution. Morerecently, much more complex devices have been presented to cope withthese problems: Yacouhian et al. (“Digital-to-analog conversion usingelectrooptic modulators,” IEEE Photonics Technology Letters, vol. 15,pp. 117-119, January 2003), proposed the employment of one MZI modulatorfor each and every bit. A recently reported design by Leven et al. (“A12.5 g samples/s optical digital-to-analog converter with 3.8 effectivebits,” Lasers and Electra-Optics Society, 2004, LEOS 2004. The 17thAnnual Meeting of the IEEE, vol. 1, pp. 270-271, November 2004), alsothe subject of U.S. Pat. No. 7,061,414 entitled “OpticalDigital-To-Analog Converter” to Y K Chen et al., employs a singlemodulator for every 2 bits and is highly nonlinear; it yields only 3.8effective bits for a 6 bit design.

There is therefore a need for a digital to analog converter which wouldoffer improved linearity of response without sacrificing efficiency ordynamic range.

SUMMARY OF THE INVENTION

The present invention is a linearized optical digital-to-analogmodulator.

According to the teachings of the present invention there is provided, amodulator device for converting digital data into analog modulation ofthe power of an optical signal, the modulator device comprising: (a) anelectronic input for receiving an input data word of N bits; (b) anelectrically controllable modulator for modulating the intensity of anoptical signal, the modulator including M actuating electrodes whereM≥N; and (c) an electrode actuating device associated with theelectronic input and the modulator, the electrode actuating device beingresponsive to the input data word to supply an actuating voltage to theactuating electrodes, wherein the electrode actuating device actuates atleast one of the actuating electrodes as a function of values of morethan one bit of the input data word.

According to a further feature of the present invention, the electrodeactuating device includes a digital-to-digital converter.

According to a further feature of the present invention, the modulatoris a modulated semiconductor light generating device. According to analternative feature of the present invention, the modulator is anelectro-absorption modulator. According to yet a further alternative,the modulator is a Mach-Zehnder modulator.

According to a further feature of the present invention, in the case ofa Mach-Zehnder modulator, the modulator includes M actuating electrodeson each of two waveguide branches of the modulator. In certain preferredcases, M is greater than N.

According to a further feature of the present invention, in the case ofa Mach-Zehnder modulator, the electrode actuating device is configuredto actuate the first and second pluralities of actuating electrodes soas to modulate the optical signal according to a QAM (QuadratureAmplitude Modulation) modulation scheme with at least 16 constellationpoints.

According to a further feature of the present invention, the electrodeactuating device is configured to actuate the first and secondpluralities of actuating electrodes so as to modulate the optical signalto a minimum amplitude for a return-to-zero signal between successiveinput data words.

According to a further feature of the present invention, the modulatorhas a maximum dynamic range, and wherein the electrode actuating deviceis configured to actuate the actuating electrodes so as to generatemodulation of the optical signal spanning a majority of the dynamicrange.

According to a further feature of the present invention, the electrodeactuating device is configured to apply one of two common actuatingvoltages to the actuating electrodes.

According to a further feature of the present invention, the actuatingelectrodes have differing effective areas. According to one set ofapplications, the differing effective areas form a set, members of theset being interrelated approximately by factors of two. In otherpreferred cases, the set including at least one effective area which isnot interrelated to others of the set by factors of two.

According to a further feature of the present invention, the modulatorhas a non-linear response, and the electrode actuating device isconfigured to actuate the actuating electrodes so as to generate animproved approximation to a linear modulation of the optical signal as afunction of the input data word.

According to a further feature of the present invention, there is alsoprovided an optical to electrical converter deployed so as to generatean electrical signal as a function of intensity of the modulated opticalsignal.

There is also provided according to a further feature of the presentinvention, an apparatus comprising a digital-to-analog converter, theconverter comprising: (a) an electronic input for receiving an inputdata word of N bits; (b) an electrically controllable modulator formodulating the intensity of an optical signal, the modulator including Mactuating electrodes where M≥N; and (c) an electrode actuating deviceassociated with the electronic input and the modulator, the electrodeactuating device being responsive to the input data word to supply anactuating voltage to the actuating electrodes, wherein the electrodeactuating device actuates at least one of the actuating electrodes as afunction of values of more than one bit of the input data word.

There is also provided according to the teachings of the presentinvention, a method for converting a digital data input word of N bitsinto an analog signal comprising: (a) processing the digital data inputword to generate an electrode actuation vector of M values where M≥N;and (b) applying M voltage values corresponding to the actuation vectorvalues to M actuating electrodes of an electrically controllablemodulator for modulating the intensity of an optical signal, wherein atleast one value of the actuation vector varies as a function of valuesof more than one bit of the input data word.

According to a further feature of the present invention, the electrodeactuation vector is a binary vector, and wherein the M voltage valuesare selected from two voltage levels according to the M binary values.

According to a further feature of the present invention, the processingis performed by a digital-to-digital converter.

According to a further feature of the present invention, an electricaloutput is generated as a function of the intensity of the modulatedoptical signal.

There is also provided according to the teachings of the presentinvention, a modulator device for converting digital data into analogmodulation of the power of an optical signal, the modulator devicecomprising: (a) an electronic input for receiving an input data word ofN hits; (b) a semiconductor light generating device for generating anoptical signal of variable intensity, the semiconductor light generatingdevice including M actuating electrodes where M≥N; and (c) an electrodeactuating device associated with the electronic input and thesemiconductor light generating device, the electrode actuating devicebeing responsive to the input data word to supply an actuating voltageto the actuating electrodes, thereby generating an output intensitycorresponding substantially to the input data word.

According to a further feature of the present invention, the actuatingelectrodes have differing effective areas.

According to a further feature of the present invention, the differingeffective areas form a set, members of the set being interrelatedapproximately by factors of two.

According to a further feature of the present invention, the differingeffective areas form a set, the set including at least one effectivearea which is not interrelated to others of the set by factors of two.

According to a further feature of the present invention, M=N.

According to a further feature of the present invention, thesemiconductor light generating device is a semiconductor laser.

According to a further feature of the present invention, thesemiconductor laser further includes a threshold electrode configured toprovide a threshold actuation current.

According to a further feature of the present invention, thesemiconductor light generating device is a light emitting diode.

There is also provided according to the teachings of the presentinvention, a modulator device for converting digital data into analogmodulation of the power of an optical signal, the modulator devicecomprising: (a) an electronic input for receiving an input data word ofN hits; (b) an electrically controllable modulator for modulating theintensity of an optical signal, the modulator including M actuatingelectrodes where M≥N; and (c) an electrode actuating device associatedwith the electronic input and the modulator, the electrode actuatingdevice being responsive to the input data word to supply an actuatingvoltage to the actuating electrodes, wherein the actuating electrodeshave differing effective areas, the differing effective areas forming aset, the set including at least one effective area which is notinterrelated to others of the set by factors of two.

At this point, it will be useful to define various terminology as usedherein in the description and claims. The terms “digital” and “analog”are used in their normal senses as common in the field. Specifically,“digital” refers to a form of data where values are stored or processednumerically, typically broken up into bits of a binary number formachine processing, whereas “analog” refers to a form of data in whichvalues are represented by different levels within a range of values ofan essentially continuously variable parameter.

The phrase “digital-to-digital converter” is used to refer to a devicewhich maps a set of possible digital input values to a set of possibledigital output values, where the input and output values arenon-identical. The “digital-to-digital converter” employed by certainembodiments of the present invention is a non-trivial converter in whichthere is typically not a one-to-one mapping between bits of the inputdata and bits of the output data, as will be clear from the descriptionfollowing.

The term “binary” is used to refer to values, voltages or otherparameters which assume one or other of only two possible values, andmodes of operation which use such parameters. In this context, voltagelevels are referred to as “common” to a number of electrodes ifactivation of the electrodes is performed by switching connection ofeach of the electrodes between the voltage values in question.

The term “electrode” is used to refer to the electrical connections ofan optical modulator device through which the device is controlled. Inthe case of an electrode which applies an electric field to affect theoptical properties of an adjacent material, reference is made to an“effective area” which is used as an indication of the relativeinfluence of the electrode compared to that of other electrodes on theoptical properties of the underlying waveguide if actuated by a similarvoltage. In many cases, the actuating electrodes are all of the sameeffective width, for example where they overlie a long narrow waveguide.The “effective area” may then be referred to as an “effective length”,corresponding to the length of waveguide overlaid by the correspondingelectrode and related to the “effective area” by a constant scalingfactor. This scaling factor will vary according to variations in shape,width, waveguide properties or other design parameters. Any part of theelectrode not overlying the active part of the modulator device orotherwise ineffective for generating modulation of an optical signal isnot included in the “effective area”.

The term “modulator” is used to refer to any device which outputs anoptical signal with controlled variation of intensity, whether thevariation is induced during production of the signal (such as in asemiconductor laser) or whether a signal input from another source ismodified.

The term “optical power” is used to refer to the quantitativemanifestation of the analog optical signal.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The invention is herein described, by way of example only, withreference to the accompanying drawings, wherein:

FIG. 1 is a schematic representation of a modulator device, constructedand operative according to the teachings of the present invention, forconverting digital data into analog modulation of an optical orelectrical signal;

FIG. 2A is a graph showing the intensity output generated by anunmodified Mach-Zehnder modulator employing four electrodes with lengthsinterrelated by factors of two and driven directly by corresponding bitsof a digital input word, the output being shown relative to a linecorresponding to an ideal linear response;

FIG. 2B is a graph showing an intensity output generated according to animplementation of the present invention employing a full-rangeMach-Zehnder modulator with four electrodes of lengths interrelated byfactors of two driven according to the teachings of the presentinvention;

FIG. 2C is a graph similar to FIG. 2B showing the intensity outputachieved by further modifying electrode lengths according to a furtheraspect of the present invention;

FIG. 3A is a graph showing the intensity output achieved by themodulator of the present invention implemented with five actuatingelectrodes for a four hit input word, where the five electrodes havelengths interrelated by factors of two;

FIG. 3B is a graph similar to FIG. 3A showing the intensity outputachieved by further modifying electrode lengths according to a furtheraspect of the present invention;

FIG. 4 is a table illustrating a digital-to-digital converter input andoutput where the input corresponds to the input data word and the outputcorresponds to the electrode actuation pattern for generating theoutputs of FIGS. 2B and 2C;

FIG. 5 is a table illustrating unoptimized and optimized normalizedelectrode lengths for cases on input words of 4 and 8 bits and numbersof electrodes of 4, 5, 8, 9 and 10;

FIG. 6A is a graph showing the intensity output generated by anunmodified Mach-Zehnder modulator employing eight electrodes withlengths interrelated by factors of two and driven directly bycorresponding bits of a digital input word, the output being shownrelative to a line corresponding to an ideal linear response;

FIG. 6B is a graph showing an intensity output generated according to animplementation of the present invention employing a Mach-Zehndermodulator with eight electrodes of lengths interrelated by factors oftwo driven according to the teachings of the present invention;

FIG. 6C is a graph similar to FIG. 6B showing the intensity outputachieved by further modifying electrode lengths according to a furtheraspect of the present invention;

FIG. 7A is a graph showing the intensity output achieved by themodulator of the present invention implemented with nine actuatingelectrodes for an eight bit input word, where the nine electrodes havelengths interrelated by factors of two;

FIG. 7B is a graph similar to FIG. 7A showing the intensity outputachieved by further modifying electrode lengths according to a furtheraspect of the present invention;

FIG. 8 is a schematic representation of an alternative implementation ofthe present invention based upon an electro-absorption modulator (EAM);

FIG. 9 is a schematic representation of yet another alternativeimplementation of the present invention based upon a semiconductorlaser;

FIG. 10 is a schematic representation of a modulator device, similar tothe device of FIG. 1 , implemented as a 16-QAM modulator;

FIG. 11A is a constellation diagram for the device of FIG. 10illustrating a choice of constellation points, and correspondingelectrode actuation patterns, for implementing a 16-QAM modulator;

FIG. 11B is a graph showing the symbol error rate performance for thedevices of FIG. 12A implemented with different numbers of electrodes;

FIGS. 12A and 12B are simplified constellation diagrams showing onlyclosest matching points for implementation of a 64-QAM using two sets of7 electrodes and a 256-QAM using two sets of 10 electrodes,respectively;

FIGS. 13A and 13B are graphs showing the symbol error rate performancefor the devices of FIGS. 12A and 12B, respectively; and

FIG. 14 is a schematic representation of a prior art Mach-Zehndermodulator configured as a DAC.

DESCRIPTION OF SPECIFIC EMBODIMENTS OF THE INVENTION

The present invention is a modulator device for converting digital datainto analog modulation of an optical signal.

The principles and operation of modulator devices according to thepresent invention may be better understood with reference to thedrawings and the accompanying description.

Referring now to the drawings, FIG. 1 shows schematically a modulatordevice, generally designated 10, constructed and operative according tothe teachings of the present invention, for converting digital data intoanalog modulation of an optical signal. Generally speaking, modulatordevice 10 has an electronic input 12 for receiving an input data word Dof N bits and an electrically controllable modulator 14 for modulatingthe intensity of an optical signal represented by arrow 16. Modulator 14includes M actuating electrodes 18 where M≥N. Modulator device 10 alsoincludes an electrode actuating device 20 responsive to the input dataword D to supply an actuating voltage to the actuating electrodes 18. Itis a particular feature of a first aspect of the present invention thatelectrode actuating device 20 actuates at least one of actuatingelectrodes 18 as a function of values of more than one bit of the inputdata word D. In other words, at least one of the electrodes is actuatedin a manner differing from a simple one-to-one mapping of data hits toelectrode voltage, thereby providing freedom to choose the electrodeactuation pattern which best approximates a desired ideal output for thegiven input. According to a second complementary, or alternative, aspectof the present invention, the effective areas of actuating electrodesare optimized so that at least some of the electrode effective areasdiffer from a simple factor-of-two series.

The basic operation of a first preferred implementation of modulatordevice 10 will be understood with reference to FIGS. 2A and 2B. FIG. 2Ashows the output intensity values which would be obtained by supplying avoltage sufficient to generate full dynamic range modulation to a set offour electrodes according to a direct mapping of each hit of a 4-bitinput data word to a corresponding electrode. The full range of inputvalues 0000 through 1111 and the corresponding output intensities havebeen normalized to the range 0-1. The marked deviation from linearity inthe form of a cosine variation is clearly visible. In contrast, FIG. 2Bshows the output intensity using the same four electrodes after theinput data has been mapped according to the teachings of the presentinvention to a pattern of electrode actuation approximating more closelyto a linear response. In other words, for each input value, the outputvalue from FIG. 2A most closely approximating the correspondingtheoretical linear response is determined, and the corresponding patternof electrode actuation is applied. By way of example, in FIG. 2A, itwill be noted that output point 22 corresponding to an input of 0011 ishigher than desired for the ideal linear response. The outputs generatedby electrode actuation patterns corresponding to value 0100 (point 24)is closer to the required value, and 0101 (point 26) is even closer. Anoutput pattern of 0101 is thus chosen to correspond to an input of 0011.The overall result is an output which much more closely approximates toa linear response as shown.

Most preferably, electrode actuating device 20 includes adigital-to-digital converter. It will be appreciated that such aconverter may be implemented from very straightforward and high-speedlogic components which make it feasible to employ the present inventionin high frequency systems. Electronic input 12 may be simply the inputpins of digital-to-digital converter 20. FIG. 4 illustrates a preferredimplementation of the digital-to-digital mapping employed to generatethe output of FIG. 2B. A digital-to-digital converter suitable forimplementing the various embodiments of the present invention mayreadily be implemented using commercially available high-speedApplication Specific Integrated Circuits (“ASIC”), as will be clear toone ordinarily skilled in the art.

The first implementation described thus far features N=M=4 with lengthsof the electrodes retaining the conventional ratios of factors of twoand employing simple on-off level voltage switching of a commonactuating voltage to all currently actuated actuating electrodes. Whilesuch an implementation offers markedly improved linearity of responsecompared to the unmodified response of FIG. 2A, it should be noted thatthe output intensity is not uniquely defined for each input value. Whereuniqueness of the output values is required, or where a higher degree oflinearity is needed, further modification may be required. Various formsof further modification may be employed including, but not limited to:use of multiple actuating voltage levels; use of modified electrodelengths; and addition of additional electrodes (i.e., M>N). Thesedifferent options will be discussed below.

One option for further modification of the output is to modify theactuating voltage applied to each electrode, such as by switchingbetween different distinct voltage levels.

An alternative preferred option for modifying the output to achieve abetter approximation to a linear output is modification of the electrodelengths relative to the factor of two series assumed above. Anon-limiting example of an approach for determining preferred electrodeproportions will be presented below in the context of a Mach-Zehndermodulator. A corresponding practical example of electrode length valuesfor N=M=4 is shown in the second column of FIG. 5 . FIG. 2C illustratesthe intensity output employing electrodes with lengths in theproportions listed. A comparison of the root-mean-square error fromlinearity for FIGS. 2B and 2C shows that the adjustment of electrodelengths results in an additional slight improvement to linearity.

A further option for modifying the performance of modulator device 10 isthe addition of one or more additional electrodes, i.e., M>N. Thisprovides an additional degree of freedom for correcting non-linearity ofthe response. In the case of unmodified electrode dimensions related byfactors of two, each additional electrode is typically half thedimension of the previously smallest electrode. Where the electrodedimensions are further modified, the additional electrode dimension ispreferably included within an optimization process in order to determinea preferred dimension for the additional electrode(s) along with theother electrodes. FIGS. 3A and 3B show outputs from the device of thepresent invention for an example of N=4 and M=5, with and withoutmodification of the electrode dimensions, respectively. The electrodelengths are shown for the unmodified series in the first column of FIG.5 and for the optimized length electrodes in the third column of FIG. 5.

Parenthetically, although the present invention is described herein inthe context of a preferred example of linearization of a modulatordevice which inherently has a non-linear response, the principles of thepresent invention may equally be applied to any case where a naturalresponse of a modulator provides a first function and a desired responseis a different second function which may be linear or non-linear. Thus,the present invention may be employed to convert a digital input into ananalog output approximating to any desired response curve within thedynamic range of the modulator. Non-limiting examples include where adesired output response curve is sinusoidal or exponential, or where itis desired to increase the resolution or “contrast” of the output withina specific range of input values.

Clearly, the present invention is not limited to applications with 4-bitdata input, and can be implemented with substantially any number of databits commensurable with other limitations of the system, such assignal-to-noise requirements. By way of example, FIGS. 6A-6C, 7A and 7Billustrate a number of implementations with 8-bit input data words.Specifically, for purpose of reference, FIG. 6A shows the unmodifiedoutput of an 8-bit arrangement where each data bit is applied directlyto a corresponding electrode, again showing the underlying cosineresponse of the modulator. FIG. 6B illustrates the output of animplementation according to the teachings of the present invention withN=M=8 and standard lengths of electrodes interrelated by factors of 2,as in the fourth column of FIG. 5 . FIG. 6C shows output for a similardevice where the electrode lengths have been modified according to thevalues shown in the fifth column of FIG. 5 . FIGS. 7A and 7B show theoutput of similar devices for N=8 but with an extra electrode, i.e.,M=9. In the case of FIG. 7A, the device has unmodified electrode lengthsas shown in the fourth column of FIG. 5 , while in the device of FIG.7B, the electrode lengths are modified according to the proportionsshown in the sixth column of FIG. 5 .

EXAMPLE I Mach-Zehnder Modulator

By way of example, there will now be presented a theoretical treatmentof one particularly preferred example of modulator device 10 implementedusing a Mach-Zehnder modulator, also referred to as a Mach-ZehnderInterferometer or “MZI”. This theoretical treatment is presented tofacilitate an understanding of the present invention and as a suggestedtechnique for calculating certain parameters. However, it should benoted that the invention as described above has been found to beeffective, independent of the accuracy or otherwise of this theoreticaltreatment.

The Mach-Zehnder modulator is an active integrated waveguide deviceconsisting of a higher index guide region that splits into two pathswhich are combined again after a certain distance. Each of these pathsis referred to as a leg or branch. When used as a switch, the MZI may beturned “off” by raising or lowering the index of refraction in one ofthe legs. This is achieved by employing the electro-optic effect toproduce a 180-degree change in phase by means of optical-path length.Intermediate optical attenuation levels can be obtained by inducingchanges other than 180 degrees.

In the case described above of FIG. 1 , a 4-bit Digital-to-AnalogConverter, based on a Multi-Electrode (ME) Mach-Zehnder Interferometer,is presented. The input to the device consists of 4 bits. Using theDigital-to-Digital converter, which may be thought of as a look-uptable, the 4 data bits are mapped to 5 electrodes as this realization isequipped with a single excess electrode. According to one option, if anelectrical rather than optical output is desired, the optical signal atthe output is detected and converted to an electrical (analog) signal.

It should be noted that, in the context of a MZI, it is common to splitthe electrodes to act in an opposite manner on the two legs of thedevice, for example, one side raising and the other lowering therefractive index of the material, thereby reducing the actuation voltagerequired. In such cases, the value M is the number of actuatingelectrodes on each of the two waveguide branches of the modulator.

Mathematical Description

The properties of the MZI may be described mathematically as follows.Let I denote a vector of electrode lengths. The number of elements in Iis M. Also let V denote a corresponding vector (of length M) ofelectrode control-voltages. When applying a voltage V_(j) only toelectrode e_(j), whose length is l_(j), the phase of light propagatingin the modulating leg, shifts by

$\pi\frac{v_{j} \cdot l_{j}}{v_{\pi} \cdot l_{\pi}}$where V_(π)·l_(π) corresponds to the voltage-length product leading to aπ shift in the phase. It is used as a merit figure of the MZI modulator.We define new normalized electrode length by:

$\begin{matrix}{L_{j} = \frac{l_{j}}{l_{\pi}}} & (1)\end{matrix}$Gathering the total contribution from all electrodes, the followingtransmission function of the MZI is obtained:

$\begin{matrix}{{{T\left( {V,L} \right)} = {\cos^{2}\left( {\frac{\pi}{2}\frac{V \cdot L^{T}}{v_{\pi}}} \right)}},} & (2)\end{matrix}$where the superscript T denotes transposition. The contribution fromeach electrode e_(j), j=1, 2, . . . M, to the total phase shift, ispermitted by applying some non-zero voltage V_(j)=v. We chose to workwith binary values for all electrode voltages, V_(j)=0, v, a clearlydesirable requirement which, moreover, makes the design simpler asdiscussed next. Note that by setting the lower voltage to a valuegreater than zero, the maximum output level of the MZI is decreased thusreducing the dynamic range. Let D_(i) denote a digital binary inputvector of length N, where i=1, . . . , 2^(N). For each digital vectorD_(i), the DDC component in FIG. 1 produces a corresponding binaryvector B_(i), of length M, B_(i) multiplied by v, represents the actual(internal) vector of voltages controlling the M electrodes. Whenmultiplying a real number, B_(i) should be interpreted as binary vectorwhose elements are the real numbers {0, 1}. With respect to (2), thismeans that when V_(j)=v, then B_(ij)=1, and when V_(j)=0, B_(ij)=0;B_(ij) being the j-th element of the control vector B_(i). At this timewe define B to be a matrix of size 2^(N)×M whose rows consist of the setof binary control vectors B_(i), each of length M. Hence, (2) can berewritten as

$\begin{matrix}{{T\left( {B_{i},L} \right)} = {\cos^{2}\left( {\frac{\pi}{2}\frac{v}{v_{\pi}}{\sum\limits_{j = 1}^{M}{B_{ij}L_{j}}}} \right)}} & (3)\end{matrix}$

Without loss of generality V=V_(π) will be assumed henceforth.(Preferably V_(j)=v_(π) to ensure full coverage of the modulating rangeand efficient use of the input optical power.) When the number ofelectrodes equals the number of data bits, i.e. when M=N, animplementation in a standard approach according to the system of FIG. 10while trying to exploit the dynamic range of the transfer function (3)to its fullest results in high nonlinearity errors. This is demonstratedin FIG. 2A for M=N=4. For this demonstration it is assumed that anunoptimized straightforward selection is made, where B is the set of all16 binary 4-tuples, i.e. B^(T)={0000, . . . , 1111} and L is taken as{0.5, 0.25, 0.125, 0.0625}. In the next section, one suggested approachto optimizing L and B will be proposed.

Optimization of B and L

In order to improve the linearity as well as the dynamic range of theconversion process, we propose that the lengths of the elements of L andthe control vectors B be optimized. As described above, it is possibleto optimize one or both of B and L. In practice, optimization of L alonemay provide a non-monotonic variation of output together with someimprovement in linearity and dynamic range. This may be sufficient forapplications in which the non-linearity is relatively small, such as thesemiconductor laser embodiments to be discussed below. For moresignificantly non-linear devices, the options of optimized B withunoptimized L, and optimized B and L are typically more suitable.

We consider these options separately since their implementation requiredifferent hardware. An unoptimized set of electrode lengths L consistsof

L_(j)=2^(−j), with j=1 . . . M. An unoptimized matrix B will consist ofall 2^(N) binary N-tuples. In that case, with a slight abuse ofnotations we have that B_(i)=D_(i); i=0,1,2, . . . 2^(N)−1. Hence,designs with optimized B require Digital-to-Digital conversion whiledesigns with optimized L only do not. Whenever B is optimized, and forany number of electrodes M; M≥N, it is understood that a binary inputdata vector D_(i) has to be mapped to a control vector B_(i), yetB_(i)≠D_(i). The DDC, implemented as all-electronic, shall perform thismapping operation.

As the optimization criterion we shall use the root mean square error(RMSE) between an ideal output, represented by a straight line, and theconverter output. Let

$U_{i} = \frac{i}{2^{N}}$denote the ideal analog value required for representing the digitalinput D_(i). The RMSE is defined as follows:

$\begin{matrix}{{g\left( {B,L} \right)} = \sqrt{\frac{1}{2^{N}}{\sum\limits_{i = 1}^{2^{N}}\left\lbrack {U_{i} - {\cos^{2}\left( {\frac{\pi}{2}{\sum\limits_{j = 1}^{M}{B_{ij}L_{j}}}} \right)}} \right\rbrack^{2}}}} & (4)\end{matrix}$

The optimization problem can now be formulated as minimizing the valuesof g(B,L) for all possible values of the matrix B and the vector L.

Note that this optimum solution is aimed at minimizing the average(squared) deviation between the desired output and the converter output.Clearly, this is only one non-limiting example, and various otherlinearity measures may equally be used. Similarly, as mentioned earlier,the desired output response function itself may take any desired form,and for each function, a suitable optimization criterion must beselected.

Approaching (4) as a global optimization problem with an order of0(2^(N)×M) variables, is quite involved, especially since the variablesare of mixed type, B is binary while L is real. (It is related to anonlinear mixed integer zero-one optimization problem.) It is thereforetypically preferred to employ a near-optimum two-step approach. First, Bis determined assuming an unoptimized set of electrodes L, L_(j)=2^(−j),with j=1 . . . M. The obtained matrix is denoted by {circumflex over(B)}. Then, given {circumflex over (B)}, L is obtained such that (4) isminimized.

If L is an unoptimized set of electrodes, L_(j)=2^(−j). Then, the outputof the converter as given by (3) is a function of the control B_(i)only. Since one aims at obtaining a straight line, whose quantizedvalues are given by U_(i), then it is not difficult to verify that thebest approximated selection of {circumflex over (B)}_(i) is given by

$\begin{matrix}{{{\hat{B}}_{i} = {{Dec}2{{Bin}_{M}\left( {\frac{2}{\pi}{\arccos\left( \sqrt{U_{i}} \right)}} \right)}}},} & (5)\end{matrix}$where the function Dec2Bin_(M)(x) maps a real value x, 0<x≤1, to itsclosest M-bit binary representation. Note that this, in effectquantization, process may result in several input data vectors havingthe same analog representation. In applications in which this duplicaterepresentation is considered problematic, it is effectively mitigated bychoosing M>N.

Given {circumflex over (B)}, we proceed to optimize L. Assuming thereexists a set of electrodes L such that

${{{\cos^{2}\left( {\frac{\pi}{2}{\sum\limits_{j = 1}^{M}{{\hat{B}}_{ij}L_{j}}}} \right)} \approx U_{i}};{\forall i}},$then as an alternative to (4), we may define an equivalent costfunction, which is easier to handle mathematically:

$\begin{matrix}{{h(L)} = {\left\{ {\sum\limits_{i = 1}^{2^{N}}\left\lbrack {{\frac{2}{\pi}{\arccos\left( \sqrt{U_{i}} \right)}} - {\sum\limits_{j = 1}^{M}{{\hat{B}}_{ij}L_{j}}}} \right\rbrack^{2}} \right\}.}} & (6)\end{matrix}$To minimize this cost function, one needs to differentiate h(L) withrespect to L and equate to 0:

$\begin{matrix}{{\frac{\partial h}{\partial L_{k}} = {{2\left\{ {\sum\limits_{i = 1}^{2^{N}}{{\hat{B}}_{ik}\left\lbrack {{\frac{2}{\pi}{\arccos\left( \sqrt{U_{i}} \right)}} - {\sum\limits_{j = 1}^{M}{{\hat{B}}_{ij}L_{j}}}} \right\rbrack}} \right\}} = 0}};{\forall{k.}}} & (7)\end{matrix}$The following equation (more precisely set of equations) is obtained

$\begin{matrix}{{{\sum\limits_{i = 1}^{2^{N}}{\sum\limits_{j = 1}^{M}{{\hat{B}}_{ik}{\hat{B}}_{ij}L_{j}}}} = {\sum\limits_{i = 1}^{2^{N}}{{\hat{B}}_{ik}\frac{2}{\pi}{\arccos\left( \sqrt{U_{i}} \right)}}}},} & (8)\end{matrix}$where k=1, 2 . . . M. In matrix notation the set of equation translatesto a simple expression

$\begin{matrix}{{L = {\left( {{\hat{B}}^{T}\hat{B}} \right)^{- 1}{\frac{2}{\pi}\left\lbrack {{\arccos\left( \sqrt{U} \right)}\hat{B}} \right\rbrack}^{T}}},} & (9)\end{matrix}$where √{square root over (U)} U amounts to a component-wise square root.The solution above grants us an optimized vector of lengths L.

EXAMPLE II Electro-Absorption Modulator

As mentioned earlier, the present invention is not limited toimplementations based on Mach-Zehnder modulators, and can be implementedusing any device which modulates light intensity as a function ofapplied voltage. By way of one additional non-limiting implementation,FIG. 8 shows an analogous implementation of the present invention usingan electro-absorption modulator.

An electro-absorption modulator (EAM) is a semiconductor device whichallows control of the intensity of a laser beam via an electric voltage.Its operational principle is typically based on the Franz-Keldysheffect, i.e., a change of the absorption spectrum caused by an appliedelectric field, which usually does not involve the excitation ofcarriers by the electric field.

By realizing N or more electrodes we can use the EAM as a high speedelectro-optical Digital-to-Analog converter, in a similar fashion as weused the MZI. As in modulator device 10 described above, this deviceincludes an electronic input 12 for receiving an input data word D of Nbits and an electrically controllable modulator 14 with M electrodes formodulating the intensity of an optical signal represented by arrow 16.An electrode actuating device 20 is responsive to the input data word Dto supply an actuating voltage to the actuating electrodes 18.

Here too, to mitigate the non-linear behavior of the device, electrodeactuating device 20 serves as a Digital-To-Digital Converter is employedto map an N bit input to a set of M electrodes, determining which of theM electrodes is actuated for each input value. The particular mappingvaries according to the response characteristic of the particularmodulator, but the principles of operation are fully analogous to thosedescribed above in the context of the Mach-Zehnder modulatorimplementation.

EXAMPLE III Modulated Light Generation Device

The present invention is applicable also to other devices where digitalinformation carried by voltage or current is translated into analogoptical signals in the form of optical power. This includes also lightgeneration devices Like Light Emitting Diodes (LED) or semiconductorlasers. By way of illustration, FIG. 9 shows a semiconductor laserimplementation of the present invention.

Specifically, FIG. 9 shows a semiconductor laser DAC device, generallydesignated 40, constructed and operative according to the teachings ofthe present invention. The actuating electrode, typically implemented asa single contiguous electrode, is here subdivided into a thresholdelectrode 42 and M actuating electrodes 18. The threshold electrode 42is typically needed to reach a minimum activation current below which nosignificant output is generated. Actuating electrodes 18 are actuated asa function of the data bits of a digital input 12, in this case a 4-bitdata word, to generate an analog optical signal output of intensityrepresenting the digital input.

In the particularly simple implementation illustrated here, neither Lnor B is optimized. In other words, each actuating electrode 18 is partof a set interrelated with effective areas in 2:1 relation, and eachelectrode is actuated as a function of corresponding single bit of theinput data word. The actuating current is typically roughly proportionalto the area of electrodes actuated. This case is in itself believed tobe patentable, and is thought to be of practical importance. Optionally,a closer approximation to a linear response can be achieved by modifyingL (electrode proportions) and/or B (by including a DDC, not shown), allaccording to the principles discussed in detail above.

It will be appreciated that a similar device may be implemented usingother semiconductor light generating devices, such as LEDs. Dependingupon the details of the device used, threshold electrode 42 may not benecessary. This and any other necessary device-specific modificationswill be self-evident to one ordinarily skilled in the art.

EXAMPLE IV OAM Transmitter

Although described above in the context of devices forintensity/amplitude modulation, it should be noted that variousembodiments of the present invention are also effective for modifyingthe phase of an optical signal, and can therefore be used as highlycompact and simple QAM (Quadrature Amplitude Modulation) modulators ortransmitters.

Specifically, referring back to FIG. 1 , it will be noted that, byconfiguring the digital-to-digital converter DDC 20 to apply differentactuation patterns to the electrodes on the two branches of the MachZehnder Modulator (MZM), modulation of the output signal phase can beachieved. Such an implementation will now be described with specificreference to FIGS. 10-13B.

Turning now to FIG. 10 , there is shown a modulator device, generallydesignated 100, implemented as a QAM modulator. Modulator device 100 isessentially similar to the device of FIG. 1 , but has been relabeled toconvey more clearly its function. Specifically, modulator device 100 hasa first plurality M₁ of actuating electrodes 18 a deployed in operativerelation to a first waveguide branch of the modulator and a secondplurality M₂ of actuating electrodes 18 b deployed in operative relationto a second waveguide branch of the modulator. In this case, M₁=M₂=5,giving a total number of actuating electrodes M=10. The electrodeactuating device is here shown as two distinct digital-to-digitalconverters 20 a and 20 b, which are configured to actuate the first andsecond pluralities of actuating electrodes 18 a and 18 b in response toa given input data word D_(i) so as to additionally modulate the phaseof the optical signal. Clearly, digital-to-digital converters 20 a and20 b can alternatively be combined into a single DDC with M=10 outputswith suitable connections to the actuating electrodes on both branchesof the waveguide.

It will be appreciated that modulator device 100 can serve as an optical16-QAM transmitter based on a single multi-electrode MZM (ME-MZM). Eachelectrode is divided into 5 segments, separately driven by two voltagesignals, 0 and V representing binary 0 and 1, respectively. The centerelectrode is a common ground for the active electrode segments. The roleof the modulator is to generate a desired M-QAM constellation which iscomposed of complex optical field values. The modulator is expected togenerate 2^(M) signals:s _(i) =r _(i) e ^(jθ) ^(l) , r _(i)>0, 0≤θ_(i)≤2π, i=1, . . . ,2^(M),  (10)

In our example of a 16-QAM with two sets of 5 electrodes, as an input,the QAM transmitter accepts an electrical 4-bit digital input word,denoted D_(i). The input word is mapped by two Digital-to-DigitalConverters (DDC) onto each of the 10 (electrode) segments, whose lengthsare the vectors L^(1,2). Each DDC outputs a 5-bit word, denoted as B_(i)¹ and B_(i) ². The output of transmitter can be written as:

$\begin{matrix}{E_{{out},i} = {{\frac{1}{\sqrt{2}}E_{in}\exp\left\{ {j2\pi{\sum\limits_{j}^{N_{1}}{B_{ij}^{1}L_{j}^{1}}}} \right\}} + {\frac{1}{\sqrt{2}}E_{in}\exp\left\{ {{- j}2\pi{\sum\limits_{j}^{N_{2}}{B_{ij}^{2}L_{j}^{2}}}} \right\}}}} & (11)\end{matrix}$where E_(in) the optical field amplitude entering the modulator and N₁,N₂ are the number of segments on each arm. The elements L_(j) ^(1,2) ∈L^(1,2) represent normalized electrode lengths on each arm. Thetwo-level B_(ij) ^(1,2) coefficients are elements of the matrices B_(i)^(1,2) and represent whether voltage v was applied to the j-th segment,on the respective arm. The index j enumerates the electrodes, j={1 . . .N_(1,2)} on each arm. The summation is normalized to span 0≤Σ_(j)^(N)B_(ij) ^(1,2)L_(j)≤1, such that each arm induces a phaseshift of0≤Δφ≤2π.The application of the electrical signals is preferably directly uponthe modulator without any mediating circuits, referred to herein as“Direct Digital Driving”. The modulator can be regarded as a 2DDigital-to-Analog (D/A) converter, that converts a digital word into anoptical vector signal.

The design of the transmitter involves the setting of electrode lengths,L^(1,2) and DDC mappings, B_(i) ^(1,2) that will generate all therequired signals given in Eq. (10). An effective combination of theelectrode lengths and digital mappings may be derived either byanalytical methods or numerically. A simple numerical derivation willnow be presented.

A ME-MZM with N_(1,2) electrode segments on each arm is capable ofgenerating 2^(N) ¹ ^(+N) ² ⁾ signals. Thus, by choosing carefully theelectrode lengths, and assuming the number of segments sufficient togenerate at least 2^(N) ¹ ^(+N) ² ⁾>M different signals, all therequired signals described by Eq. (10) can be picked out of thegenerated signals pool.

As an example, FIG. 11A shows the an ideal Square-16-QAM constellation,which is the required signal constellation, and a signal pool.

As an example, FIG. 11A shows the an ideal Square-16-QAM constellation,which is the required signal constellation, and a signal pool which wasgenerated with {5,5} electrodes. The best matched signals at the poolare marked in figure. It can be seen that there is a good match betweenthe ideal and the best matched generated constellations.

Table 1 compares between an ideal 16-QAM constellation and a generatedconstellation with different combinations of number of electrodes eacharm. It presents the symbol minimum distance and the root mean squareerror. The latter provides a measure of agreement between the ideal andthe generated constellations. Configurations with {2,2}, {2,3} and {3,3}electrodes provide less than 16 different signals (minimum distance of0) and therefore cannot be used for generation of 16-QAM.

TABLE 1 N₁, N₂ Ideal 4, 2 4, 3 4, 4 4, 5 5, 5 5, 6 6, 6 Minimum 2 1.661.66 1.66 1.30 1.83 1.67 1.67 Distance RMSE 0 4.64 4.64 4.53 4.42 4.644.52 4.57

FIG. 11B presents the Symbol Error Rate (SER) performance for a range ofSignal to Noise Ratios (SNR) for an Additive White Gaussian Noise (AWGN)channel. It can be seen that when using {6,6} electrodes, theperformance graph closely matches that of ideal 16-QAM constellation.Using a higher number of electrodes can lower the SER ever furthertoward the ideal curve. SER performance can be slightly improved byfurther tuning the decoding hypothesis testing at the receiver side tothe generated constellation.

The electrode lengths used for the generation of FIG. 11B follow abinary sequences, L^(1,2)=2^(−j). Tweaking the electrode lengths hassmall impact on the modulator performance. In Eq. (11), we assumeddriving signals of 0 and 2V_(π). The high driving signal can be loweredby extending the total electrode length twice its size. The oppositesigns in the exponents in the two terms of Eq. (11) are alreadyimplemented by the geometry of the electrode disposition of FIG. 10 ,when the signs of the voltages applied are the same, since then theelectric fields have opposite directions. Other electrode dispositionsexist for different cuts of the electro-optic crystals, and theappropriate way of applying the voltage with the tight signs should beclear to a person skilled in the art.

Referring now to FIGS. 12A-13B, it will be appreciated that the proposedmodulator is capable of generating high order QAM constellations. FIG.12A depicts the ideal 64-QAM constellation together with one generatedusing a {7,7} electrodes modulator, while FIG. 12B shows an ideal256-QAM constellation together with the one generated using a {10,10}electrodes modulator. It can be seen that there is a good match betweenthe ideal and the generated constellations. FIGS. 13A and 13B show theSymbol Error Rate performance for the generated constellations of FIGS.12A and 12B, respectively.

A simple implementation of this embodiment described thus far generatesNon-Return-to-Zero (NRZ) signals. NRZ permits constant intensity forsimilar consecutive hits, and is thus more susceptible toInter-Symbol-Interference and other nonlinear propagation distortions.Return-to-Zero (RZ) format is a pulsed modulation where the signal“returns to zero” after every hit. This format provides betterperformance than NRZ, but usually requires additional hardware, such asa pulse carver. A transmitter based on the modulator of an embodiment ofthe invention can readily be extended to produce RZ pulses with minimalif any additional hardware. By adding an RZ control line to the DDC, asshown in FIG. 10 , which serves as a trigger determining whether theoutput optical amplitude should be zero or other, the whole modulatorwill be capable of generating RZ signals. When the added control line ishigh, the DDC will map the electrode actuation pattern to the patterncorresponding to the middle point of the constellation, which is zero.

For the constellation presented in FIG. 10 , B_(i)={00010} andB₂={01110} will output zero (or minimum) power because it will generatea phase difference of π between the two arms of the MZM.

While the present invention has been presented as a digital-to-analogoptical modulator, it should be noted that each embodiment of theinvention may be modified to provide analog electrical output by use ofan optical-to-electrical (O/E) converter. This option is illustrated inFIG. 1 as optional O/E converter 30. If the O/E converter itself has anon-linear response function, the present invention may advantageouslybe used to optimize the system parameters to linearize the electricaloutput rather than the (intermediate) optical output. Analogously, anynonlinearity induced by the optical medium used for transmitting thesignal (e.g. optical fiber) can be compensated for by using O/Econverter 30 as a linearizing device.

The present invention is applicable to substantially all applicationsrequiring a DAC with optical or electrical output. Examples ofparticular interest include, but are not limited to, wirelesscommunications systems, fiber-optic communication systems, cellulartelephone networks, cable television, military applications, medicalapplications and hyper/super computer communications.

It will be appreciated that the above descriptions are intended only toserve as examples, and that many other embodiments are possible withinthe scope of the present invention as defined in the appended claims.

All publications, patents and patent applications mentioned in thisspecification are herein incorporated in their entirety by referenceinto the specification, to the same extent as if each individualpublication, patent or patent application was specifically andindividually indicated to be incorporated herein by reference. Inaddition, citation or identification of any reference in thisapplication shall not be construed as an admission that such referenceis available as prior art to the present invention. To the extent thatsection headings are used, they should not be construed as necessarilylimiting. In addition, any priority document(s) of this applicationis/are hereby incorporated herein by reference in its/their entirety.

The invention claimed is:
 1. An optical modulation system, the systemcomprising: an input for receiving digital input words, each digitalinput word comprising a plurality of input data bits; an input opticalsignal, wherein the optical modulation system, based on a mapping, mapsa digital input word received at the input to a corresponding digitaloutput word comprising a plurality of output data bits, wherein themapping specifies, for each digital input word included in a set ofpossible digital input words for the plurality of input data bits, acorresponding digital output word from a set of possible digital outputwords for the plurality of output data bits, wherein, for a first subsetof digital input words specified in the mapping and having increasingvalues, deltas between values of digital output words in the set ofpossible digital output words, corresponding respectively to digitalinput words in the first subset, decrease, wherein, for a second subsetof digital input words specified in the mapping and having increasingvalues, deltas between values of digital output words in the set ofpossible digital output words, corresponding respectively to digitalinput words in the second subset, increase, and wherein the opticalmodulation system further comprises a modulator for modulating the inputoptical signal responsively to the digital output word mapped from thedigital input word, thereby generating one or more modulated opticalsignal outputs for transmission over one or more optical fibers.
 2. Theoptical modulation system of claim 1, wherein at least some of thedigital input words in the first subset or the second subset specifiedin the mapping are non-contiguous.
 3. The optical modulation system ofclaim 1: wherein, for a third subset of digital input words specified inthe mapping and having decreasing values, deltas between values ofdigital output words in the set of possible digital output words,corresponding respectively to digital input words in the third subset,decrease, and wherein, for a fourth subset of digital input wordsspecified in the mapping and having decreasing values, deltas betweenvalues of digital output words in the set of possible digital outputwords, corresponding respectively to digital input words in the fourthsubset, increase.
 4. The optical modulation system of claim 3, whereinat least some of the digital input words the third subset or the fourthsubset specified in the mapping are non-contiguous.
 5. The opticalmodulation system of claim 1, wherein, for a fifth subset of increasingor decreasing digital input words in the mapping, deltas between valuesof digital output words in the set of possible digital output words,corresponding respectively to digital input words in the fifth subset,remain the same.
 6. The optical modulation system of claim 5, wherein atleast some of the digital input words in the fifth subset specified inthe mapping are non-contiguous.
 7. The optical modulation system ofclaim 1, wherein, for at least two different digital input wordsincluded in the set of possible digital input words, the correspondingdigital output words are the same.
 8. The optical modulation system ofclaim 1, wherein the modulator actuates at least one actuating electroderesponsive to the digital output word.
 9. The optical modulation systemof claim 1, wherein the one or more modulated optical signal outputscomprise analog modulated optical signals.
 10. A method, comprising:receiving at least one digital input word at an input, each digitalinput word comprising a plurality of input data bits; receiving an inputoptical signal, determining, based on a mapping, a digital output wordcomprising a plurality of output data bits mapped from a digital inputword received at the input, wherein the mapping specifies, for eachdigital input word included in a set of possible digital input words forthe plurality of input data bits, a corresponding digital output wordfrom a set of possible digital output words for the plurality of outputdata bits, wherein, for a first subset of digital input words specifiedin the mapping and having increasing values, deltas between values ofdigital output words in the set of possible digital output words,corresponding respectively to digital input words in the first subset,decrease, and wherein, for a second subset of digital input wordsspecified in the mapping and having increasing values, deltas betweenvalues of digital output words in the set of possible digital outputwords, corresponding respectively to digital input words in the secondsubset, increase; and modulating the input optical signal responsivelyto the digital output word mapped from the digital input word; therebygenerating one or more modulated optical signal outputs for transmissionover one or more optical fibers.
 11. The method of claim 10, wherein atleast some of the digital input words in the first subset or the secondsubset specified in the mapping are non-contiguous.
 12. The method ofclaim 10: wherein, for a third subset of digital input words specifiedin the mapping and having decreasing values, deltas between values ofdigital output words in the set of possible digital output words,corresponding respectively to digital input words in the third subset,decrease, and wherein, for a fourth subset of digital input wordsspecified in the mapping and having decreasing values, deltas betweenvalues of digital output words in the set of possible digital outputwords, corresponding respectively to digital input words in the fourthsubset, increase.
 13. The method of claim 12, wherein at least some ofthe digital input words the third subset or the fourth subset specifiedin the mapping are non-contiguous.
 14. The method of claim 10, wherein,for a fifth subset of increasing or decreasing digital input words inthe mapping, deltas between values of digital output words in the set ofpossible digital output words, corresponding respectively to digitalinput words in the fifth subset, remain the same.
 15. The method ofclaim 14, wherein at least some of the digital input words in the fifthsubset specified in the mapping are non-contiguous.
 16. The method ofclaim 10, wherein, for at least two different digital input wordsincluded in the set of possible digital input words, the correspondingdigital output words are the same.
 17. The method of claim 10, furthercomprising actuating at least one actuating electrode responsive to thedigital output word.
 18. The method of claim 10, wherein the one or moremodulated optical signal outputs comprise analog modulated opticalsignals.
 19. An optical modulation system, the system comprising: aninput for receiving digital input words, each digital input wordcomprising a plurality of input data bits; an input optical signal,wherein the optical modulation system, based on a mapping, maps adigital input word received at the input to a corresponding digitaloutput word comprising a plurality of output data bits, wherein, for afirst plurality of digital input words applied to the mapping and havingincreasing values, values of digital output words, correspondingrespectively to the first plurality of digital input words, decrease,wherein, for a second plurality of digital input words applied to themapping and having increasing values, values of digital output words,corresponding respectively to the second plurality of digital inputwords, increase, and wherein the optical modulation system furthercomprises a modulator for modulating the input optical signalresponsively to the digital output word mapped from the digital inputword, thereby generating one or more modulated optical signal outputsfor transmission over one or more optical fibers.
 20. The opticalmodulation system of claim 19, wherein at least some of the firstplurality of digital input words specified in the mapping arenon-contiguous.
 21. The optical modulation system of claim 19, whereinat least some of the second plurality of digital input words specifiedin the mapping are non-contiguous.
 22. The optical modulation system ofclaim 19: wherein, for a third plurality of digital input words appliedto the mapping and having decreasing values, values of digital outputwords, corresponding respectively to the third plurality of digitalinput words, decrease, and wherein, for a fourth plurality of digitalinput words applied to the mapping and having decreasing values, valuesof digital output words, corresponding respectively to the fourthplurality of digital input words, increase.
 23. The optical modulationsystem of claim 22, wherein at least some of the third plurality ofdigital input words or the fourth plurality of digital input wordsspecified in the mapping are non-contiguous.
 24. The optical modulationsystem of claim 19, wherein, for a fifth plurality of digital inputwords applied to the mapping and having increasing or decreasing values,values of digital output words, corresponding respectively to the fifthplurality of digital input words, remain the same.
 25. The opticalmodulation system of claim 24, wherein at least some of the fifthplurality of digital input words specified in the mapping arenon-contiguous.
 26. The optical modulation system of claim 19, whereinthe modulator actuates at least one actuating electrode responsive tothe digital output word.
 27. The optical modulation system of claim 19,wherein the one or more modulated optical signal outputs comprise analogmodulated optical signals.
 28. A method, comprising: receiving at leastone digital input word at an input, each digital input word comprising aplurality of input data bits; receiving an input optical signal,determining, based on a mapping, a digital output word comprising aplurality of output data bits mapped from a digital input word receivedat the input, wherein, for a first plurality of digital input wordsapplied to the mapping and having increasing values, values of digitaloutput words, corresponding respectively to the first plurality ofdigital input words, decrease, wherein, for a second plurality ofdigital input words applied to the mapping and having increasing values,values of digital output words, corresponding respectively to the secondplurality of digital input words, increase; and modulating the inputoptical signal responsively to the digital output word mapped from thedigital input word, thereby generating one or more modulated opticalsignal outputs for transmission over one or more optical fibers.
 29. Themethod of claim 28, wherein at least some of the first plurality ofdigital input words specified in the mapping are non-contiguous.
 30. Themethod of claim 28, wherein at least some of the second plurality ofdigital input words specified in the mapping are non-contiguous.
 31. Themethod of claim 28: wherein, for a third plurality of digital inputwords applied to the mapping and having decreasing values, values ofdigital output words, corresponding respectively to the third pluralityof digital input words, decrease, and wherein, for a fourth plurality ofdigital input words applied to the mapping and having decreasing values,values of digital output words, corresponding respectively to the fourthplurality of digital input words, increase.
 32. The method of claim 31,wherein at least some of the third plurality of digital input words orthe fourth plurality of digital input words specified in the mapping arenon-contiguous.
 33. The method of claim 28, wherein, for a fifthplurality of digital input words applied to the mapping and havingincreasing or decreasing values, values of digital output words,corresponding respectively to the fifth plurality of digital inputwords, remain the same.
 34. The method of claim 33, wherein at leastsome of the fifth plurality of digital input words specified in themapping are non-contiguous.
 35. The method of claim 28, furthercomprising actuating at least one actuating electrode responsive to thedigital output word.
 36. The method of claim 28, wherein the one or moremodulated optical signal outputs comprise analog modulated opticalsignals.
 37. An optical modulation system, the system comprising: aninput for receiving digital input words, each digital input wordcomprising a plurality of input data bits; an input optical signal,wherein the optical modulation system maps a digital input word receivedat the input to a corresponding digital output word comprising aplurality of output data bits, wherein, for a first plurality of digitalinput words received successively at the digital input and havingincreasing values, deltas between values of digital output words mappedrespectively from the first plurality of digital input words decrease,wherein, for a second plurality of digital input words receivedsuccessively at the digital input and having increasing values, deltasbetween digital output words mapped respectively from the secondplurality of digital input words increase, and wherein the opticalmodulation system further comprises a modulator for modulating the inputoptical signal responsively to the digital output word mapped from thedigital input word, thereby generating one or more modulated opticalsignal outputs for transmission over one or more optical fibers.
 38. Theoptical modulation system of claim 37: wherein, for a third plurality ofdigital input words received successively at the digital input andhaving decreasing values, deltas between values of digital output wordsmapped respectively from the third plurality of digital input wordsdecrease, and wherein, for a fourth plurality of digital input wordsreceived successively at the digital input and having decreasing values,deltas between values of digital output words mapped respectively fromthe fourth plurality of digital input words increase.
 39. The opticalmodulation system of claim 37, wherein, for a fifth plurality of digitalinput words received successively at the digital input and havingdecreasing or increasing values, deltas between values of digital outputwords mapped respectively from the fifth plurality of digital inputwords remain the same.
 40. The optical modulation system of claim 37,wherein the modulator actuates at least one actuating electroderesponsive to the digital output word.
 41. The optical modulation systemof claim 37, wherein the one or more modulated optical signal outputscomprise analog modulated optical signals.
 42. A method, comprising:receiving at least one digital input word at an input, each digitalinput word comprising a plurality of input data bits; receiving an inputoptical signal, mapping a digital input word received at the input to acorresponding digital output word comprising a plurality of output databits, wherein, for a first plurality of digital input words receivedsuccessively at the digital input and having increasing values, deltasbetween values of digital output words mapped respectively from thefirst plurality of digital input words decrease, wherein, for a secondplurality of digital input words received successively at the digitalinput and having increasing values, deltas between digital output wordsmapped respectively from the second plurality of digital input wordsincrease; and modulating the input optical signal responsively to thedigital output word mapped from the digital input word, therebygenerating one or more modulated optical signal outputs for transmissionover one or more optical fibers.
 43. The method of claim 42: wherein,for a third plurality of digital input words received successively atthe digital input and having decreasing values, deltas between values ofdigital output words mapped respectively from the third plurality ofdigital input words decrease, and wherein, for a fourth plurality ofdigital input words received successively at the digital input andhaving decreasing values, deltas between values of digital output wordsmapped respectively from the fourth plurality of digital input wordsincrease.
 44. The method of claim 42, wherein, for a fifth plurality ofdigital input words received successively at the digital input andhaving decreasing or increasing values, deltas between values of digitaloutput words mapped respectively from the fifth plurality of digitalinput words remain the same.
 45. The method of claim 42, furthercomprising actuating at least one actuating electrode responsive to thedigital output word.
 46. The method of claim 42, wherein the one or oremodulated optical signal outputs comprise analog modulated opticalsignals.
 47. An optical modulation system, the system comprising: aninput for receiving digital input words, each digital input wordcomprising a plurality of input data bits; an input optical signal,wherein the optical modulation system maps a digital input word receivedat the input to a corresponding digital output word comprising aplurality of output data bits, wherein, for a first plurality of digitalinput words received successively at the digital input and havingincreasing values, deltas between values of digital output words mappedrespectively from the first plurality of digital input words decrease,wherein, for a second plurality of digital input words receivedsuccessively at the digital input and having increasing values, deltasbetween digital output words mapped respectively from the secondplurality of digital input words increase, wherein, for a thirdplurality of digital input words received successively at the digitalinput and having decreasing values, deltas between values of digitaloutput words mapped respectively from the third plurality of digitalinput words decrease, wherein, for a fourth plurality of digital inputwords received successively at the digital input and having decreasingvalues, deltas between values of digital output words mappedrespectively from the fourth plurality of digital input words increase,wherein, for a fifth plurality of digital input words receivedsuccessively at the digital input and having decreasing or increasingvalues, deltas between values of digital output words mappedrespectively from the fifth plurality of digital input words remain thesame, and wherein the optical modulation system further comprises amodulator for modulating the input optical signal responsively to thedigital output word mapped from the digital input word, therebygenerating one or more modulated optical signal outputs for transmissionover one or more optical fibers.
 48. The optical modulation system ofclaim 47, wherein the modulator actuates at least one actuatingelectrode responsive to the digital output word.
 49. The opticalmodulation system of claim 47, wherein the one or more modulated opticalsignal outputs comprise analog modulated optical signals.
 50. A method,comprising: receiving at least one digital input word at an input, eachdigital input word comprising a plurality of input data bits; receivingan input optical signal, mapping a digital input word received at theinput to a corresponding digital output word comprising a plurality ofoutput data bits, wherein, for a first plurality of digital input wordsreceived successively at the digital input and having increasing values,deltas between values of digital output words mapped respectively fromthe first plurality of digital input words decrease, wherein; for asecond plurality of digital input words received successively at thedigital input and having increasing values, deltas between digitaloutput words mapped respectively from the second plurality of digitalinput words increase, wherein, for a third plurality of digital inputwords received successively at the digital input and having decreasingvalues; deltas between values of digital output words mappedrespectively from the third plurality of digital input words decrease,wherein, for a fourth plurality of digital input words receivedsuccessively at the digital input and having decreasing values, deltasbetween values of digital output words mapped respectively from thefourth plurality of digital input words increase, and wherein, for afifth plurality of digital input words received successively at thedigital input and having decreasing or increasing values, deltas betweenvalues of digital output words mapped respectively from the fifthplurality of digital input words remain the same; and modulating theinput optical signal responsively to the digital output word mapped fromthe digital input word; thereby generating one or more modulated opticalsignal outputs for transmission over one or more optical fibers.
 51. Themethod of 50, wherein the modulator actuates at least one actuatingelectrode responsive to the digital output word.
 52. The method of 50,wherein the one or more modulated optical signal outputs comprise analogmodulated optical signals.
 53. An optical modulation system, the systemcomprising: an input for receiving digital input words, each digitalinput word comprising a plurality of input data bits; an input opticalsignal, wherein the optical modulation system, based on a mapping, mapsa digital input word received at the input to a corresponding digitaloutput word comprising a plurality of output data bits, wherein themapping specifies, for each digital input word included in a set ofpossible digital input words for the plurality input data bits, acorresponding digital output word from a set of possible digital outputwords for the plurality of output data bits, wherein, for a first subsetof digital input words specified in the mapping and having increasingvalues, deltas between values of digital output words in the set ofdigital output words, corresponding respectively to digital input wordsin the first subset, decrease, wherein, for a second subset of digitalinput words specified in the mapping and having increasing values,deltas between values of digital output words in the set of digitaloutput words, corresponding respectively to digital input words in thesecond subset, increase, wherein, for a third subset of digital inputwords specified in the mapping and having decreasing values, deltasbetween values of digital output words in the set of digital outputwords, corresponding respectively to digital input words in the thirdsubset, decrease, wherein, for a fourth subset of digital input wordsspecified in the mapping and having decreasing values, deltas betweenvalues of digital output words in the set of digital output words,corresponding respectively to digital input words in the fourth subset,increase, wherein, for a fifth subset of digital input words specifiedin the mapping and having increasing or decreasing values, deltasbetween values of digital output words in the set of digital outputwords, corresponding respectively to digital input words in the fifthsubset, remain the same, wherein for any of the first, second, third,fourth, or fifth subsets, at least some of the digital input wordswithin the subset specified in the mapping are non-contiguous, andwherein the optical modulation system further comprises a modulator formodulating the input optical signal responsively to the digital outputword mapped from the digital input word, thereby generating one or moremodulated optical signal outputs for transmission over one or moreoptical fibers.
 54. The optical modulation system of claim 53, whereinthe modulator actuates at least one actuating electrode responsive tothe digital output word.
 55. The optical modulation system of claim 53,wherein the one or more modulated optical signal outputs comprise analogmodulated optical signals.
 56. A method, comprising: receiving at leastone digital input word at an input, each digital input word comprising aplurality of input data bits; receiving an input optical signal,determining, based on a mapping, a digital output word comprising aplurality of output data bits corresponding to a digital input wordreceived at the input, wherein the mapping specifies, for each digitalinput word included in a set of possible digital input words for theplurality input data bits, a corresponding digital output word from aset of possible digital output words for the plurality of output databits, wherein, for a first subset of digital input words specified inthe mapping and having increasing values, deltas between values ofdigital output words in the set of digital output words, correspondingrespectively to digital input words in the first subset, decrease,wherein, for a second subset of digital input words specified in themapping and having increasing values, deltas between values of digitaloutput words in the set of digital output words, correspondingrespectively to digital input words in the second subset, increase,wherein, for a third subset of digital input words specified in themapping and having decreasing values, deltas between values of digitaloutput words in the set of digital output words, correspondingrespectively to digital input words in the third subset, decrease,wherein, for a fourth subset of digital input words specified in themapping and having decreasing values, deltas between values of digitaloutput words in the set of digital output words, correspondingrespectively to digital input words in the fourth subset, increase,wherein, for a fifth subset of digital input words specified in themapping and having increasing or decreasing values, deltas betweenvalues of digital output words in the set of digital output words,corresponding respectively to digital input words in the fifth subset,remain the same, and wherein for any of the first, second, third,fourth, or fifth subsets, at least some of the digital input wordswithin the subset specified in the mapping are non-contiguous; andmodulating the input optical signal responsively to the digital outputword mapped from the digital input word, thereby generating one or moremodulated optical signal outputs for transmission over one or moreoptical fibers.
 57. The method of claim 56, wherein the modulatoractuates at least one actuating electrode responsive to the digitaloutput word.
 58. The method of claim 56, wherein the one or moremodulated optical signal outputs comprise analog modulated opticalsignals.